Satoru Shoji

21-104, Wakayama-dai, Shimamoto, Osaka, Japan

Keywords:Jump variability Coupled vibration In-bore balloting Cantilever tube

ABSTRACT The variability of projectile jump is the long-term issue to improve weapon accuracy.Nowadays we have many simulation codes to predict Jump. However, these codes cannot explain explicitly how the variability of jump arises.The aims of this paper are,1)to give fundamental explanations for the variability of Jump, and 2) to offer design factors to make the variability of Jump less. The model presented here was formulated in accordance with the transition of the system in reverse way commencing from the target impact stage to the chambering stage. Objects of the simulation are generic 120 mm smooth bore and long rod system which are extremely simplified to the vibration of tube, the spring effect by sabot, and the free-ends beam of penetrator. Parametric calculations clarified that high Jump-variability is generated only when the last rebound is on the muzzle line. This particular state of rebound is achieved by many combinations of input-variable. Guide map named JV-Chart is proposed to show the high Jumpvariability zone.

1. Introduction

Projectile jump (?jump for short), is the sight angle difference between the calculated aiming point and the actual impact point.The serious problem of jump is that the angle difference is not stable and we cannot fix the offset for the FCS(fire control system)of a tank. We conventionally call this unstable jump “occasion-tooccasion variation” or “shot-to-shot dispersion”, and we call them“jump-variability” in this paper.

We have two aims in this paper. One is to fully understand the mechanism of jump-variability, and the second is to find design guides to decrease jump-variability. Nowadays many simulation software have been developed but those are ineffective for investigating jump-variability because,

a. They do not point out the fundamental cause of jumpvariability.

b. They have no assured grounds for assigning input elements.

Taking into account the background of jump investigation, we will build a new jump model that can predict and explain the jumpvariability.

2. Jump simulation model

2.1. Scope of modelling

1) Simulation was limited to two dimensional motion,and we have neglected some muzzle exit effects such as uneven pressure,gaseous explosion, or irregular discard of sabot.

2) A generic 120 mm smooth bore gun and long rod projectile were taken for a simulation object.

3) Appendix A: “Symbols and Data” shows main default values,symbols and configuration for convenience of reference. The physical unit system used in this paper is “gram-millimetermillisecond” unless otherwise noted.

4) “Rebound” means either the name of motion or magnitude of force of the rebound. “Input-variable” is the input or cause which is prepared for parametric calculation.

5) Appendix B: “Flow of Variables”shows the outline of our jump simulation model.

2.2. Exterior flight

Standard exterior ballistics does not calculate jump.Fortunately,forerunners have already formulated the jump.The three attributes of a body (mass-point, rigid-body, and continuum) take the independent actions in air stream. Projectile jump (? jump), Jp is the summation of three independent jumps. (Eq. (1)). The velocity jump, Jv is the motion of point mass which is proportional to the lateral velocity vZof rod (Eq. (2)). The rigid-body jump (? Bundy jump), JBcomes from the aerodynamic effect on rigid body under rotation around center of mass. The angular velocity of the rod at the muzzle, uZwill start with asymmetrical yawing which eventually results in the lateral shift in trajectory(Eq.(3)).This jump is formulated by Bundy[1].The elastic-body jump(?Schmidt jump),Js is also the aerodynamic effect on the continuum beam.This effect comes from the difference of air drag at forward half and the rear half of the flexure rod(Eq.(4)).This jump is formulated by Schmidt[2].

The variables vZ,uZ;and dZof the above three formulae will be born from the collisions between tube and projectile. We will call these collisions as “In-Bore Balloting”.

2.3. In-bore balloting of rod

Our simulation model is,in essence,“the coupling vibration of a rod”in a tube.The sabot in which a rod is held is accelerated by gas pressure and makes contacts to and bounces off the deflecting tube many times.This balloting motion continues until exiting from the muzzle.All in-bore calculations will be done in“bore coordinates”that is defined as an orthogonal frame travelling along the vibrating center-line of the gun tube.

Dynamic properties of a rod at the muzzle will be given by Eq.(5),Eq.(6)and Eq.(7).yc is the transverse distance of rod center to tube center,and sc is the inclination of rod relative to the rod center as seen in Fig. 1. The second terms in Eq. (5) and Eq. (6) are the lateral velocity and angular velocity of tube center line at the time of muzzle exit respectively. dZof Eq. (7) is the first maximum deflection of rod after launch.This equation was deduced from the extreme condition that the deflection velocity of the rod center reaches zero.yDc is the deflection of rod shown at Eq.(16).Dot over X means the partial derivative of X with respect to time.

Fig.1. The 2DOF coupling vibration.

2.3.1. Equation of motion of rod as rigid body

The rod in bore can be described as“the two degree of freedom(2DOF) non-linear coupling vibration”. The constitution of the vibration model is shown in Fig.1.The lateral transition yc[t]and the inclination sc[t ] against tube centerline (? bore coordinates axis)can be determined by solving the equations of motion,Eq. (8) and Eq.(9),respectively.And the second terms of Eq.(8)and Eq.(9)are the translational acceleration of the bore coordinate and angular acceleration of the bore coordinate, respectively.

Compression forces(or rebound forces),fRAof Eq.(10)and fRFof Eq.(11)are generated between rod and tube along the lines of action“A”and “F”.“A” is the position of aft-bore rider(or rear bulkhead)and “F” is the position of forward bore rider (or front bell).

Sabot is assumed to act as compression springs with no mass.kA and kF are the spring constants that work between convex sabot and concave tube as seen in Fig. 2, the proportionality of compression was confirmed with the static load test by Lyon [3].

dAand dFare the distances between tube center and rod center at“A” and “F” respectively.

Cs is the compensating rate of the spring constant which varies with the orientation ? of the sabot petal (see Fig. 3). “0.7” is the ratio of minimum to maximum compression which was obtained experimentally by Lyon[3].The function Eq.(14)was formulated so as to change the compression smoothly as the orientation changes(see Fig. 3).

Function reBND(Eq.(15))was introduced to correct the rebound force so as to evaluate the effect of the radial clearance(hoA or hoF)between tube and rod.H[x]is the Heaviside step function.Fig.4 is the graphical expression of reBND.

Fig. 2. The compression of sabot.

Fig. 3. Orientation of sabot at loading.

2.3.2. Flexure of rod

The rod, as a homogeneous cylindrical beam without any support, will vibrate making flexure yDC?x; the in accordance with the Euler-Bernoulli formula under the transverse load intensities given by Eq. (18). The symbol “x” is the independent variable of position along the rod.

where,

Fig. 4. Installation of clearance: hoA or hoF by way of function reBND.

The convolution method(?Duhamel integral)was modified for use of discrete variables to the lateral load intensities on rod as shown in Eq.(18).sDC; Eq.(19),is the tangent of the flexure rod at the position x and time t.This equation has no direct relation to the jump variability though,rod end angle(as seen in Fig.14),sDC?LC;the is an attractive motion which we can measure directly with mirror and laser-beam technique.

2.4. Gun tube deflection

The gun tube is supported by a tube carriage of tank to be a sliding cantilever. The vibration of the gun tube changes the boundary of the sabot motion and generates lateral acceleration of the bore coordinates.In order to simulate typical droop curve,tube was supposed as a hollow cylinder with four step-wise outer diameters as illustrated in Fig. 5.

Two imaginary tubes were formulated in order to see the effect of their asymmetry on projectile jump.Using these equations is not mandatory (see Eq. (20) and Eq. (21)), experimentally taken data are also useable. Two types of tube, droop type (Fig. 6) and warp type(Fig.7),were taken into consideration.The initial curvature of the center line of the imaginary gun tubes are shown in the illustrations below.Vertical scales are written in mm.Horizontal scales are the distance from RFT (rear face of tube) in mm.

The tube vibrates in accordance with Euler-Bernoulli theory as a sliding cantilever supported by a carriage. A mass eccentricity of tube against breech centerline produces a sudden inertial moment upon recoil at every position.The total induced moment of force is converted to transverse load intensity on the tube that was formulated as Eq.(24)by Simkins[4].Lateral motion of the carriage is supposed to be zero until the exit of projectile from the muzzle.One more load on the tube is the projectile load due to the rebound of projectile against tube, which is formulated as Eq. (25).

For the “droop” type tube with uneven outer diameter.

Fig. 5. Cantilever gun tube.

Fig. 6. Center line of the Droop type tube - Eq. (20)

Fig. 7. Center line of the warp type tube - Eq. (21).

Fig. 8. Deflection of droop-type tube.

Fig. 9. Deflection of warp-type tube.

For the “warp” type tube; (bbw is the half width of the warp).This proposed function-Eq.(21)has the benefits;1)zero shear force at the top and the slope-ends which represent natural deformation,2) the property of addition which can simulate any shape of deformation.

where,

Fig.10. dA and dF at “A” and “F” respectively.

Fig.11. Rebound forces fRA and fRF. Vertical line designates the muzzle position. During rebound, sabot is sliding on the tube-inside surface.

Fig.12. Deviation (left, in mm) and lateral velocity (right, in mm/ms) of rod center.

Recoil acceleration aR?[t]is determined from the conservation of momentum which is held among the axially moving parts of the system.“0.5” is the usually used rough value.

The axial acceleration of projectile aP?[t]is given by Eq.(27)as a hypothetical interior ballistics data for the convenience of parametric change of data.

where,pss0 is the shot-start pressure,and other three constants,a1,a2, and a3are fixed by giving maximum travel LZ, time to exit tZ,maximum pressure pmax, and muzzle velocity v0.

Eq. (28) is the tangent of the deflecting tube at position z and time t.Then sDT?[LT;tZ]will be the centerline angle of the muzzle at the time of projectile exit.

3. Results

In this section, some numerical results are shown to demonstrate the various aspects of in-bore balloting. Mathematical equations are coded on a computer algebra system (CAS).

3.1. Deflection of gun tube

Figs. 8 and 9 below demonstrate the deflection of the tube in mm. The moving curves are overlaid as the projectile travels by 60 mm each increment.

Fig.13. Inclination (left in radian) and angular velocity (right in rad/ms) around rod center of mass.

Fig.14. Effects of the damping coefficient ehCT on rod flexure.

Fig.15. Jump curves (Jp[hcF] and Jp[kF]).

Fig.16. Progressive shift of rebound position by clearance (hcF).

3.2. Output from single shot

The following graphs are the typical outputs from a single shot taken for the purpose of demonstrating the in-bore balloting.

Oscillatory motions of dAand dF, are the distances between the tube center and rod center at “A” and “F” respectively. The two symmetrically drawn horizontal-lines in Fig.10 indicate the range where rod can move freely.For the left graph,horizontal lines at“A”is, and for the right hand graph, horizontal lines at “F” is. Areas beyond these lines are proportional to the “rebound force”.

Fig.17. Degressive shift of rebound by sabot-stiffness (kF).

Fig.18. Three-D Display of JP [hcF, hcA].

Fig.19. (a) 3D display of JV [hcF, hcA]. (b) JV-Chart of JV [hcF, hcA].

Fig. 11 shows the calculated rebound forces. The right side vertical line in both figures indicates the position of the muzzle.During rebound,sabot is sliding on the tube inside-surface.Figs.12 and 13 show the motions of rod in the moving bore coordinate system.

The inclinations of rod end angle (Fig.14) are calculated by Eq.(19).It was found that the end angles are sensitive to the damping coefficient hCof rod vibration.This damping coefficient comes from the internal friction of rod material.“hC”appears at the last formula of Equation (16).

3.3. Parametric change of jump

Fig. 15 displays the calculated jumps. Shot numbers are given from left to right as one increment of input.Connected 21 points in each graph designate the magnitude of jumps in milli-radian. It takes 4 min to calculate one shot with a customary personal computer.At the steep portion of each curve,a small change of variable makes a big change in jump.It is obvious that 0.02 mm of change in the radial play makes 1.0 mrad of difference in jump when the average (or designed) clearance is around 0.1 mm. From the practical point of view, this change of jump is tremendously large and cannot be accepted. There are many variety of pattern of jumpcurves depending on the type of input-variable.

Fig.16.(2)is a good example where the steep slope of the curve of Fig.15 corresponds to the halfway cut-off of the last rebound.The similar correspondence is seen also in Fig.17.(2).On the other hand,

Table 1 Input variables.

the flat portions of jump curve which are designated by four black arrows in Fig. 15 correspond to the full form (no cut-off) of the rebound force.

3.4. From JP function to JV-Chart

3.4.1. JP solid figure (jump surface)

When the two input-varaibles (hcA, hcF) are changed parametrically at the same time, the calculated jump function JP[hcA,hcF] will make 3-D surface (Fig.18).

3.4.2. JV chart (jump variability chart)

We have defined a generalized slope Eq. (29) for 3-D curved surface. The jump function JP (Fig. 18) can be converted to jump variability JV (Fig.19a) through this formula.to show high variability zone(s).The coordinates scales of this chart are not the actual value of input,but are indices(e.g.shot number;j or k) used for the calculation of JP.

3.4.3. The effect of the third variable

The JV-Chart of JV [hcF, hcA] of Fig. 19b is a function of two variables. When the input-variable kF (spring constant along “F”line)was joined as the third variable,JV[hcF,hcA,kF]changes as kF increases as seen in Fig. 20.

3.5. JV-charts from a pair of input-variables

The combinations of two input-variables are calculated to make JV-Charts. Each calculation range is divided into ten points. The parametric points are tabulated in Table 1, (k?1,2, …,11).

Case 1: Combinations of two input-variables at a time selected from projectile properties (see Fig. 21).

Note that JV has the dimension of “radian/increment”, that means, the addition of the different type of JP has no physical meaning but has the practical merrit to normalize the jumpvariability at the input coordinates(y1, y2).

Fig.19b is the top view of Fig.19a.The greenish blue transparent mesh plane was set at 0.3 mrad as a temporary acceptance level for convenient viewing.JV values above the flat mesh looks like islands in an ocean or hills in a plain;then we call this picture a“JV-Chart”

Fig.21 will give us the advices about design of the span between two riders and of the position of rod relative to the aft rider position.

Case 2: Combinations of two input-variables at a time selected from each of projectile and tube.The caption Y-X means vertical Y and horizontal X like the same way as Case 1.

On the Charts of the Y-aaw group (latter six charts in Fig. 22),there seems a tendency that more than 0.25 mm of “warp height:aaw” will be the cause of high variability in the system of the current default values.On the Y-ccw case(peceding six in Fig.22),it shall not be construed as being no general tendency, because the coordinates for “ccw” are taken far wider range(1700 mme3700 mm)compared to the narrow range of variables of projectiles.Such big change of ccw is nothing to do with“Shot-toshot dispersion”, but will make “Occasion-to-Occasion Variation”.

Fig. 21. JV-charts of 6C2 (?15) combinations from six factors.

Fig. 22. Jump variability by warp of tube: [Twelve JV-Charts].

Fig. 23. Jumps of serial aaw.

Fig. 24. Jumps of serial ccw.

Fig. 25. JP solid figure ofcombined aaw and ccw. Green flat plane is the zero level of jump.This graph represents the jumps of 441(?21×21)shots which include the data of Figs. 23 and 24.

Fig.26. JV-Chart of gun tube. Data of Fig.25 are converted by the conversion formula Eq. (29). The horizontal red line corresponds to the input-varaible aaw of Fig. 23,whereas the vertical red line corresponds to the input-varaible ccw of Fig. 24.Threshold plane was set at 0.3 mrad level. Regions which stand out above the green plane indicate higher jump variability resion.

Designers both of gun and projectile must have, before their start of design,some arrangements on the principal input-variables of the system based on the common understandings of the jumpvariability.

Case 3: Jump Variabilities from tube.

The input-variables aaw and ccw which are attributed to tube only can generate high variability of jump. From Fig. 26, it can be seen that the warp-center near the muzzle will make high variability.

4. Discussions

4.1. Causes of jump variability

Fig. 27. RML-state.

The width of rebound is the sliding distance of sabot under compression. High variability of jump will take place at the state where the last rebound occupies the muzzle position. We call this state “RML-state” (Fig. 27) in this paper. The RML-state has the following significant features

a. RML-state works as an amplifier of jump

b. RML-state is attained by particular combinations of inputvaraibles

c. RML-states agglomerate themselves because of their inter

mittant nature.

There are two methods to identify the input-variables which direct to the RML-state.

One approach is to make JV-Chart as was described in Section 3.4.Another approach is to solve the RML-state functional-equations with respect to input-variables.This method is not yet in our hands but is the challenging, attractive issue.

4.2. Explanations on unique phenomena

Now we have possible explanations for the unique phenomena of jump. We had ignored the amplification characteristics of inbore balloting. And this amplification works only when the system is under a specific condition called RML-state. This is the reason why we could not identify the causes of shot-to-shot dispersion for a long time.

The followings are the possible causes of high variability of jump. The magnitude of them is dependent on the engineering of manufacturer of the individual system,though.

a. Clearances between parts especially in radial direction;

b. Natural deflection (as warp) of tube;

c. Insufficient air purge in working oil;

d. Degree of wear at the forcing cone or unstable shot-start pressure;

e. Transformation of material induced by the shock of live firing.

4.3. JV-chart and its usage

JV-Chart is a simplest tool to know the conditions where the high variability of jump will be.However,it is not practical to try all combinations of input-variables. Then the second best is to make JV-Chart ofNC2combinations. N types of input shall be chosen by engineering judgement. The acceptable combinations of input values must be in the area where the green part is common to all charts. If we cannot find a common green area, then we have to change the default value(s)of input or shall make trade-off among input-variables.The last option we have to choose is the downgrade of the acceptance level of variability.

Understanding the Jump-variability must be the basis for the overall design of a gun-projectile system that includes planning of experiment, quality control, design and logistics.

4.4. Further development

We have brought up the present Jump-Variability model by the name of“EUBON2D”.We expect the following developments of the present model together with the experimental tests.

1) Taking into the bobbing motion of MBT as an “elastic foundation” of cantilevered tube.

2) Algebraic research to formulate and solve the“RML-state”with respect to all inputs.

5. Conclusions

a. The coupling vibration model can explain many unique phenomena in live firing.

b. High jump-variability arises with the particular combinations of inputs.

c. The JV-Chart was proposed as a tool for finding the conditions of the high variability of jump.

d. The model fabrication from outputs to inputs was found effective to look into the nature.

Appendix A. Symbols and data

Fig. A.1. System Configurations.

Appendix B. Flow of variables